([0,1]-adapted MRA) We define the
spaces
where the sign
refers to the
-orthogonality.

Proposition

The spaces
,
and
are mutually
-orthogonal.

Proof

The statement follows from construction: the collection
is
-orthogonal
and the spaces
,
and
are linear span of
for different values of the index
.

We would like to find an
-orthogonal
basis for the space
.
The functions
are already
-orthogonal.
Before we perform Gram-Schmidt orthogonalization of
and
we need to establish that both collections are linearly independent.

Proposition

(Restricted linear independence)
The collection
is linearly independent on
.

(Maximal dimension 1) The collection
is linearly independent. The collection
is linearly independent.

Proof

We introduce the
notation
We aim to establish that the collection
is linearly independent then the statement would follow from the proposition
(
Restricted linear
independence
).

We
have
We make the change
,
,
.
where
and, according to the proposition
(
Integral of scaling
function
),
Thus
and linear independence of
follows from linear independence of
.

We proceed with orthogonalization. We seek the functions
such
that
We substitute the first relationship into the
second:
The
is a square symmetric positive-definite matrix. Hence, there exists a Choleski
decomposition:
and the
choice
is sufficient to produce
-orthogonal
collection
.
The collection
is
-orthogonal
to
because it is a linear combination of
.

Proposition

(Resolution
structure for adapted scale functions) The spaces
have the
structure

Proof

We saw in the proof of the proposition
(
Maximal dimension 1
) that there exist
two-way linear
transformations
where

Due to the conditions
is suffices to show that
may be represented as a linear combination of
.

We
calculate
We use the proposition (
Scaling
equation
).
We use the formula (
Property of
scale and transport
7
).
Here
,
,
hence the index
varies over
.
We intend to make a change
,
.
Thus
can only be even. We separate even and odd values of
.
For
even we have
,
,
hence, the
becomes
:
For
odd we have
,
,
:
We make a change
,
in the first sum and
in the second sum and put these sums together because then
would vary over
:
We
have
We
continue
At this point we invoke the proposition
(
Sufficient conditions
for vanishing
moments
)-d:
We separate even and odd values of
:
Note that
and
thus the proposition
(
Sufficient conditions
for vanishing moments
)-d applies for all values of
and
We
continue
The expression
is within the linear combination of
.